Multiple Comparison Procedures

1. Introduction Analysis of variance is used to test for the real treatment differences. When the null hypothesis that all the treatment means are equal is not rejected, it may seem that no further questions need to be asked. However, in some experimental situations, it may be an oversimplification of the problem. For example, consider an experiment in rice weed control with 15 treatments viz. 4 with hand weeding, 10 with herbicides and 1 with no weeding (control). The probable questions that may be raised and the specific mean comparisons that can provide their answers may be: i) Is any treatment effective in controlling the weeds? This question may be answered simply by comparing the mean of the control treatment with the mean of each of the 14 weed-control treatments. ii) Is there any difference between the group of hand-weeding treatments and the group of herbicide treatments? The comparison of the combined mean of the four hand weeding treatment effects with the combined mean of the 10-herbicide treatment effects may be able to answer the above question. iii) Are there differences between the 4 hand weeding treatments? To answer this question one should test the significant differences among the 4 hand weeding treatments. Similar question can be raised about the 10-herbicide treatments and can be answered in the above fashion. This illustrates the diversity in the types of treatment effects comparisons. Broadly speaking, these comparisons can be classified either as Pair Comparison or Group Comparison. In pair comparisons, we compare the treatment effects pairwise whereas in-group comparisons, the comparisons could be between group comparisons, within group comparisons, trend comparisons, and factorial comparisons. In the above example, question (i) is the example of the pair comparisons and question (ii) illustrates the between group comparison and question (iii) is within group comparison. Through trend comparisons, we can test the functional relationship (linear, quadratic, cubic, etc.) between treatment means and treatment levels using orthogonal polynomials. Factorial comparisons are related to the testing of means of levels of a factor averaged over levels of all other factors or average of treatment combinations of some factors averaged over all levels of other factors. For pairwise treatment comparisons there are many test procedures, however, for the group comparisons, the most commonly used test procedure is to partition the treatment sum of squares into meaningful comparisons. This can be done through contrast analysis either using single degrees …